数理解析研究所講究録 第 76850巻 1991 年 50-61
On the Classification of Locally Hamming Distance-Regular Graphs
BY MAKOTO MATSUMOTO
Abstract. A distance-regular graph is locally Hamming if it is locally isomorphic to a Hamming scheme $H(r, 2)$ . This paper rediscovers the connection among locally Ham- ming distance-regular graphs, designs, and multiply transitive permutation groups, through which we classify some of locally Hamming distance-transitive graphs.
\S 1. Introduction. By a graph we shall mean a finite undirected graph with no loops and no multiple edges. For a graph $G,$ $V(G)$ denotes the vertex set and $E(G)$ denotes the edge set
of $G$ . For a vertex $v$ of a graph $G,$ $N(v)$ denotes the set of adjacent vertices with
$v$ . By $\ovalbox{\tt\small REJECT}_{2}$ we denote the two-element field, and by $H(r)$ we denote the r-dimensional Hamming scheme for $r\geq 1$ ; that is, $H(r)$ is such a graph that its vertex set is the
$\#_{2}^{=r}$ vector space a,nd $\tau\iota,$ $v\in\#_{2}^{=r}$ are adjacent if and only if the Hamming distance $d(u, v)=1$ ; i.e., $\#\{i u_{i}\neq v_{i}\}=1$ where $u=(u_{1}, \ldots u_{r})$ and $v=(v_{1}, \ldots , v_{r})$ . The graph obtained from $H(r)$ by identifying antipodal points is called a folded Hamming sheme(or a folded Hamming cube in [3, p.140]). The graph $H(3)$ is called a cube, and the induced subgraph obtained by removing one vertex tog$e$ ther with the three incident edges from a cube is called a tulip. A tulip contains exactly three vertices with degree two, and these vertices are called petals. The unique vertex which is not adjacent to any petals is called the root of the tulip. A locally Hamming graph $G$ is a connected graph such that
(1) $G$ has no triangle,
(2) for any $u,$ $v\in V(G)$ satisfying $d(u, v)=2$ , there exist exactly two vertices
adjacent to both $u$ and $v$ ,
$T$ $G$ (3) for any subgraph of which is isomorphic to a tulip with petals $p,$ $q,$ $r$ ,
$r$ there exists a vertex $x\in V(G)$ adjacent to all $p,$ $q$ ) and . (The uniqueness of
$x$ follows from the condition (2).) Of course $H(r)$ is an example of locally Hamming graphs, and it was proved that a distance-regular graph with $parc\urcorner metersa_{1}=0,$ $a_{2}=0,$ $c_{2}=2$ , and $c_{3}=3$ is a locally Hamming $graph[2]$ [$3$ , Lemma 4.3.5]. For a generalization to $H(r,q)$ for $q\geq 3$ , see [10]. (A locally Hamming graph is exactly the same thing with a rectagraph such that any 3-claw determines a unique 3-cube in Brouwer’s terminology[3, p.153].) Our objective is to classify all locally Hamming distance-regular graphs(LHDRG). A number of authors have contributed to this $aim[2][3][5][10]$ . This paper provides an approach to this goal using a universal covering of a LHDRG. Main result is as follows.
Typeset by $A_{\mathcal{M}}S- T_{FX}$ 51
MAIN RESULT. Let $G$ be an r-regular $distaJ1ce- l\cdot eg$ ular graph with parameters $a_{1}=$ $a_{2}=0$ and $c_{2}=2,$ $c_{3}=3,$ $ot1_{1}ertl_{la1l}H(r)$ . $Tal_{\overline{1}}e$ th$emi_{1l}imun3nunIbert$ such that either $a_{t}\neq 0$ or $c_{t}\neq t$ occurs. Pu $td=2t+1$ if $c_{t}=t$ , an $d$ put $d=2tott1$erwise. Then, $tAere$ exists a $t-(r, d, /\backslash )$ design witli $/\backslash \leq(r-t)/(d-t)$ . if $G$ is distance-transitive,
$s$ $\lfloor(d-1)/2\rfloor$ $d$ $Aut(G)$ contain a -homogeneous $p_{\supset}^{\circ_{loup}}$ of egre$er$ acting on the block set of the $t-(r, d, /\backslash )$ design. COROLLARY. (See Theorem 3). Any distance-transitive graph with $a_{1}=a_{2}=a_{3}=0$ and $c_{i}=i$ for $i=1,2,3,4$ is a Hamming scheme or a folded Hamm$ingsch$eme.
For the case $t=2,3$ , some nontrivial examples $are$ listed. \S 2. The universal covering. Let $G,$ $H$ be graphs. A mapping $f$ : $V(H)arrow V(G)$ is said to be a covering if it is surjective and for any $u\in V(H),$ $f_{N(u)}$ is a bijection $N(u)arrow N(f(u))$ . For a vertex $u$ of $G$ , the set $f^{-1}(u)$ is called the fiber on $u$ . In the previous paper[9] we proved the following propositions.
PROPOSITION 1. Let $G$ be a locally $Ha$mming graph with valen$cyr$ . Then, there exists a covering $f$ : $H(r)arrow C_{\tau}$ . Let $H$ be a $loc$ally Hamming graph an $d$ le $th$ : $Harrow G$ be a covering. For any vertex $u\in V(G)$ and for any vertices $x\in f^{-1}(u),$ $y\in h^{-1}(u)$ , tliere exists a unique covering $g:H(r)arrow H$ such that $g:xrightarrow y$ and $f=hg$ hold.
The above covering $f$ is called a universal covering. We define the fundamental group $\pi(G, f)$ of the pair $G$ and $f$ : $H(r)arrow G$ as
$\pi(G, f)=\{\gamma\in Aut(H(r))|f\gamma=f\}$ .
This definition depends on the choice of $f$ , but unique upto conjugacy in $Aut(H(r))$ . (Note that this definition is not equivalent to the fundamental group of $G$ as a one- dimensional topological object.) /
Let $\Gamma$ be a subgroup of $Aut(H(r))$ . We define the discreteness $d_{\Gamma}$ of I’ by
$d_{\Gamma}= \min\{d(u,\gamma u)|\gamma\in\Gamma,\gamma\neq id, u\in V(H)\}$ where $d$ denotes the Hamming distance. For the trivial group {id}, its discreteness
$\Gamma$ is defined to be $\infty$ . Let be a subgroup of $Aut(H(r))$ with $d_{\Gamma}\geq 5$ . Then we define the quotient graph $H(r)/\Gamma$ as follows. The vertex set is the set of coset of $V(H(r))$ by $\Gamma$ ; i.e., the set $\{\Gamma u|u\in H(r)\}$ , where $\Gamma u=\{\gamma u|\gamma\in\Gamma\}$ . The two vertices $\Gamma u,$ $\Gamma v$ are adjacent if and only if there exists a vertex $w\in\Gamma u$ such that $w$ is adjacent with
$v$ . The obtained graph was proved to be locally Hamming. The canonical mapping $f$ : $H(r)arrow H(r)/\Gamma$ defined by $f$ : $u\mapsto\Gamma u$ is a universal covering. (For a detailed proof, see [9].) 52
$G$ $f$ PROPOSITION 2. Let be a locally Hamming graph with valency $r_{i}$ and let : $H(r)arrow G$ be a universal covering. Then, $d_{\pi(G,f)}\geq 5$ and $G\cong H(r)/\pi(G, f)$ hold.
Thus, the classification problem of LHDRG with valency $r$ is converted to the problem to determine for which subgroup $\Gamma$ of $Aut(H(r)),$ $H(r)/\Gamma$ is a distance- regular graph. REMARK 1: Proposition 1 was proved by Brouwer[2][3, p.153], without emphasis on the universality. Almost equivalent propositions to Propositions 1 and 2 will also be found in the book by Brouwer et. a1.[3]. What is new in this paper is that we regard a covering $f$ not only as a partition but as a coset by a group $\Gamma$ .
NOTE: In this paper, the letter $d$ always denotes not the diameter but the discrete- ness. \S 3. Designs. Recall that a $t-(v, k, \lambda)$ design is a family $B$ of k-element subset of a v-element set $X$ such that for any k-element subset $K$ of $X$ , the number of $B\in B$ containing $K$ is A. In Lemmas 3 and 3*, we shall prove a LHDRG produces a design with an additional property. To begin with, we classify LHDRG using the discreteness of its fundamental group.
Let $LHDRG_{d}^{r}$ denote the set of LHDRGs with valency $r$ such that the discreteness of its fundamental group is equal to $d$ . Later it will be proved that a distance- regular graph belongs to $LHDRG_{d}^{r}$ for some $d\geq 7$ if and only if its parameters satisfy
$a_{1}=0,$ $a_{2}=0,$ $c_{2}=2$ , and $c_{3}=3$ . Thus, the classification of LHDRG $rd$ implies the classification of distance-regular graphs with such parameters. The reason why the cases $d=5,6$ are also considered is that some interesting LHDRG with $d=5,6$ obtained from Golay codes exist, as shown in Section 5. The next lemma is useful to shorten some proofs.
LEMMA 1. Let $H(r)$ be an r-dimensional Hamming scheme and let $\Gamma$ be $a$ subgroup of
$d_{\Gamma}\geq 5$ $\Gamma x_{1},$ $\Gamma x_{2},$ $\Gamma x_{t}$ $H/\Gamma;i.e.,$ $\Gamma x_{j}$ $Aut(H(r))$ with . Let $\ldots$ , be a walk in is adjacent
to $\Gamma x_{j+1}$ for $j=1,2,$ $\ldots t-1$ . Then, there exist $x_{2}’$ , . . . , $x_{t}’suclo$ that $\Gamma x_{j}’=\Gamma x_{j}$ for
$j=2,$ $\ldots t$ and that $x_{1},x_{2}’$ , . . . , $x_{t}’$ is a walk in H. Consequently,
$d_{H(r)/\Gamma}(\Gamma x, \Gamma y)=d_{H(r)}(x, \Gamma y)$
holds, where $RHS$ denotes the minimum of $d_{H(r)}(x, \gamma y)$ for all $\gamma\in\Gamma$ .
$\Gamma x_{1}$ $\Gamma x_{2}$ $x_{2}’\in\Gamma x_{2}$ PROOF: Since is adjacent to , there exists an adjacent to $x_{1}$ .
Thus, the existence of $x_{j}’$ is inductively proved. Then there exists a $\gamma\in\Gamma$ such that $”=\gamma y$ . Thus, $d_{H(r)/\Gamma}(\Gamma x, \Gamma y)\geq d_{H(r)}(x,\gamma y)$ . Converse inequality follows because
$x$ $\Gamma y$ $f$ $\blacksquare$ a path connecting with $\gamma y$ is mapped to a path connecting I $x$ to by .
The behavior of parameters of $LHDRG_{d}^{r}$ slightly changes according to the parity of
$d$ $G$ . For a graph and its vertex $v,$ $N_{t}(v)$ denotes the induced subgraph of $G$ by all
vertices at distance at $mo$st $t$ from $v$ . 53
LEMMA 2. $Su$ppose that $G$ belongs to $LHDRG_{2t+1}^{r}$ , let $\Gamma$ be its fundamental $gro$up, and identify $G$ with $H(r)/\Gamma$ . Take an arbitrary $vertexv\in V(G)$ and $tal_{\overline{1}}e$ a vertex $u$ in the fiber $f^{-1}(v)$ . Then, $f$ embeds $1V_{t}(n)$ into $N_{i}(v)$ , and $f$ : $V(1V_{t}(u))arrow V(N_{t}(v))$ is surjective.
PROOF: Since $f$ is a covering, it is sufficient to prove that $f$ induces a bijection $V(N_{t}(u))arrow V(N_{t}(v))$ . Take $\Gamma x\in V(N_{t}(v))$ . Since $d(\Gamma u, \Gamma x)\leq t$ , Lemma 1 shows that $d(u, \Gamma x)\leq t$ , and thus there exists a vertex in $\Gamma x$ at distance $t$ from $u$ . This shows surjectivity. For the injectivity, take $x,$ $y\in N_{t}(u)$ such that $f(x)=f(y)$ ; i.e., $\Gamma x=\Gamma y$ . Then $x=\gamma y$ for some $\gamma\in\Gamma$ , and from $d_{\Gamma}=2t+1>t+t\geq d(x, u)+d(u, y)\geq d(x, y)=d(\gamma y, y)$ ,
$\gamma=id$ follows. Thus $x=y$ holds, and injectivity is proved. $\blacksquare$ For an r-element set $X$ , we can naturally identify the vertex set of $H(r)$ with the power set $2^{X}=\mathcal{P}(X)$ . The symbol $\emptyset$ denotes the empty set, which is identified with a vertex of $H(r)$ . We denote the family of d-element subset of $X$ by $(_{d}^{X})$ . We denote
$b_{i},$ by $a_{i},$ $c_{i}$ the usual parameters of a distance-regular graph $G(precise$ definition will be found in [1] or [3]).
LEMMA 3. Let $G=H(r)/\Gamma$ be a graph in $LHDRG_{2t+1}^{r}$ . Then its parameters satisf.$v$ equalities $a_{i}=0,$ $c_{i}=i$ for $i=1,2,$ $\ldots$ , $t-1$ and $a_{t}=(t+1)\lambda,$ $c_{t}=t$ for a some positive in $teger/\backslash$ . Identify the vertex set of $H(r)$ with the $po\iota ver$ se $t\mathcal{P}(X)$ for $X=\{1,2, \ldots , r\}$ . Then, the se $t\Gamma\emptyset\cap(_{2t+^{-}1}\lambda)$ consists of the block set $B$ of a $t-(r,2t+1, \lambda)$ design, with an additional property that for any $B,$ $B’\in B$ , either $B=B’$ or $\#(B\cap B’)\leq t$ holds.
$b_{i},$ PROOF: Identify $H(r)$ with $P(X)$ . The parameters $a_{i},$ $c_{i}$ for $i\leq t-1$ are de- termined immediately from Lemma 2 and the fact that $f$ is a covering. We shall fix
$\Gamma\emptyset$ $ta1_{t}’e$ the ve $r$ tex as the one end to calculate the parameters. To determine $c_{t},$ a vertex of $G$ at distance $t$ from $\Gamma\emptyset$ . By Lemma 1, this vertex can be written as $\Gamma v$ with
$d_{H(\gamma\cdot)}(\emptyset, v)=t$; i.e., with $v\in(_{t}^{X})$ . Let $u$ be a vertex of $H(r)$ adjacent with $v$ . Then, ei- ther $d_{H(r)}(\emptyset,u)=t-1$ or $t+1$ holds. The number of such $u\in N(v)$ that $d_{G}(\Gamma\emptyset, \Gamma c\iota)=$
$b_{t},$ $t,$ $t+1,$ $t-1$ , respectively, is by definition the number $a_{t},$ $c_{t}$ . First we determine $c_{t}$ . Suppose that $d_{G}(\Gamma\emptyset, \Gamma u)=t-1$ holds. Then, by Lelnma 1, $d_{H(r)}(\emptyset,\gamma u)=t-1$ for a
$\gamma\in\Gamma$ . Consequently, $d_{H(r)}(u, \gamma u)\leq d_{H(r)}(u, \emptyset)+d_{H(r)}(\emptyset,\gamma u)\leq(t+1)+(t-1) $d_{H(r)}(\gamma\emptyset, u)=t$. By considering $u$ as a $(t+1)$-element set, the latter equality is 54 equivalent to that $u$ is contained in $\gamma\emptyset$ . Conversely, if $v\in N(v)$ with $\#(u)=t+1$ is contained in $\gamma\emptyset$ with $\#(\gamma\emptyset)=2t+1$ , then $t\leq d_{G}(\Gamma\emptyset, \Gamma u)\leq d_{H(r)}(\wedge[\emptyset, u)=t$ holds. $u\in N(v)\cap(_{t+}\gamma\emptyset_{1})$ $\gamma\in\Gamma$ $\gamma\emptyset\in(_{2t+1}X)$ Thus, $a_{t}$ is the number of for some with . Suppose that such $u$ is contained in two different $\gamma\emptyset,$ $\gamma’\emptyset$ at distance $2t+1$ from 0. Since $u\subset\gamma\emptyset\cap\gamma’\emptyset,$ $\#(\gamma\emptyset\cap\gamma’\emptyset)\geq\#(u)=t+1$ holds. Then, $2t+1=d_{\Gamma}\leq$ $d_{H(r)}(\gamma\emptyset, \gamma’\emptyset)=\#(\gamma\emptyset)+\#(\gamma’\emptyset)-2\neq(\gamma\emptyset\cap\gamma’\emptyset)\leq 2t$ , a contradiction. Thus, any such $u$ is contained in at most one $\gamma\emptyset$ . Let $/\backslash _{v}$ be the number of $\gamma\emptyset\in(_{2t+1}X)$ containing $v$ . Then, above argument asserts that the number of $u\in N(v)\cap(_{t+}\gamma\emptyset_{1})$ for some $\gamma\emptyset\in(_{2t+1}X)$ is exactly $\lambda_{v}(t+1)$ , because the number of such $u$ contained a fixed $\gamma\emptyset$ is $d_{H(r)}(v,\gamma\emptyset)=t+1$ . Since $G$ is distance-regular, the value $a_{t}=/\backslash _{v}(t+1)$ does not depend on the choice of $v$ , and this implies that the set $\{\gamma\emptyset|\#(\gamma\emptyset)=2t+1, \gamma\in\Gamma\}$ consists of the block set 3 of $t-(r,2t+1, /\backslash )$ design with an additional property that for any two different $B,$ $B’\in B,$ $\#(B\cap B‘)$ $\leq t$ holds. (The last inequality follows from $d_{H(r)}(B, B’)\geq 2t+1$ for $B\neq B’.$ ) $\blacksquare$ In the case of even discreteness $d$ , the following lemmas hold. LEMMA 2*. Suppose th a $tG$ belongs to $LHDRG_{2t}^{r}$ , le $t\Gamma$ be its fundam$ental$ group, and $id$entifv $G$ with $H(r)/\Gamma$ . Take aIl arbi $t$ rary vertex $v\in V(G)$ and $t$ alre a $ver$tex $u$ in the fiber $f^{-1}(v)$ . Then, $1V_{t-1}(v)\cong N_{t-1}(u)$ holds. PROOF: This lemma can be proved in exactly the same way as in the proof of Lemma 2, by substituting $N_{t-1}$ for $N_{t}$ and putting $dr=2t$ . $\blacksquare$ LEMMA 3*. Let $G=H(r)/\Gamma$ be a graph in $LHDRG_{2t}^{r}$ . Then its paramet$ers$ satisfy equalities $a_{i}=0,$ $c_{i}=i$ for $i=1,2,$ $\ldots$ , $t-1$ and $c_{t}=t(\lambda+1)$ for some positive integer A. $Id$entify the vertex se $f$ of $H(r)$ with the power se $t\mathcal{P}(X)$ . Then, the se $t$ $\Gamma\emptyset\cap(_{2t}^{X})$ consis $6s$ of the block se $t$ Z3 of a $t-(r,2t, \lambda)d$esign, with an additional property that for any $B,$ $B’\in \mathcal{B}$ , either $B=B’$ or $B\cap B’\leq t$ hold. PROOF: From Lemma 2*, it is obvious that for $i=1,2,$ $\ldots t-2$ , the parameters $b_{t-1},$ $tal distance $t-1$ from $\emptyset$ , and let $u$ be one of its adjacent vertex. Then, we see that the number of such $u$ that $d_{H(r)}(u, \Gamma\emptyset)=t-2,$ $t-1,t$ , respectively, is $c_{t-1},$ $a_{t-1},$ $b_{t-1}$ . $\emptyset$ $\Gamma\emptyset$ The nearest element in $\Gamma\emptyset$ from $v$ is, however, only , and the other elements in are at least at distance $t+1$ from $v$ . This implies that $a_{t-1},$ $b_{t-1},$ $c_{t-1}$ also coincide with $H(r)$ . Take a vertex $v$ of $H(r)$ at distance $t$ from $\emptyset$ , and let $u$ be one of its adjacent vertices. Then, the number of such $u$ that $d_{H(r)}(u, \Gamma\emptyset)=t-1$ corresponds to $c_{t}$ . If $d(u, \Gamma\emptyset)=t-1$ then either $d(\emptyset, u)=t-1$ or $d(u,\gamma\emptyset)=t-1$ holds for some $\gamma\in\Gamma$ with $\gamma\emptyset\in(_{2t}^{X})$ . In the latter case, $\#(u)=t+1$ and $u\subset\gamma\emptyset$ holds. Any such $u$ provides an edge of c-type. Such $u$ is easily proved to be contained at most one $\gamma\emptyset\in(_{2t}^{X})$ , otherwise $d(\gamma\emptyset,\gamma’\emptyset) number of $\gamma\emptyset\in(_{2t}^{X})$ containing $v$ . Then the c-type edges incident with $v$ are exactly ones in the spans between $v$ and such $\gamma\emptyset$ or $\emptyset$ . Thus, $c_{t}=t(\lambda+1)$ for a number A 55 independent of tlre choice of $v$ , since $G$ is assumed to be distance-regular. Clearly $\Gamma\emptyset\cap(_{2t}^{X})$ consists of the block set of a $t-(\uparrow)2t_{/}\backslash )$ design. $\blacksquare$ Thus we have a slogan. SLOGAN 1. If all $t-(r, cl_{?’}\backslash )$ designs with $d=2t$ or $d=2t+1$ with an additional property $tl_{l}$ a $t$ the intersection of any two $diffel\cdot ent$ blocks $B,$ $B’$ is of size no $moI^{\cdot}e$ than $t$ are classifed, then LHDRG will be classified. Such nontrivial designs do exist. Some examples are shown in Section 5. We will use the next criterion in the next section. CRITERION 1. Let 3 be the block set of a design as in Slogan 1. Then, $/\backslash \leq(\uparrow-t)/(d-t)$ holds. PROOF: Let $X$ be the v-element set. As shown in Lemmas 3 and 3*, every $(t+1)-$ element subset of $X$ is contained in at most one block. Since one block contains $(_{t+^{d}1})$ of $(t+1)$-element subsets, the inequality $(_{t+^{r}1})\geq b(t+^{d}1)$ follows, where $b$ denotes the cardinality of the $1\supset lock$ set $\mathcal{B}$ . Combining with a well-known $identity/\backslash (tr)=b(td)$ , we have the desired inequality. $\blacksquare$ The next lemma was proved in [3, p.153] in a different terminology. LEMMA 4. A distance-regular graph $G$ belongs to LHDRGr for some $d\geq 7$ if and only if the parameters of $G$ satisfy $a_{1}=0,$ $a_{2}=0,$ $c_{2}=2$ , and $c_{3}=3$ . PROOF: The necessity immediately follows from Lemmas 3 and 3*. To prove the sufficiency, let $G$ be a distance-regular graph with the above parameters. It is sufficient to prove that $G$ satisfies the three condition of the definition of locally Hamming, since $d_{\Gamma}\geq 7$ follows from Lemmas 3 and 3*. The condition (1) follows &om $a_{1}=0$ . The $T$ $G$ condition (2) follows from $c_{2}=0$ . To prove (3), take a tulip in , let $p,$ $q,$ $r$ be its petals, and let $v$ be its root. From the conditions (2) and (1), we have a vertex $x$ in $G$ which is adjacent with both $p$ and $q$ . It is easily checked that $x$ is at distance 3 from $v$ . Since $c_{3}=3$ , there exists a vertex $y$ different from both $p$ and $q$ such that a path $x,$ $y,$ $z,$ $v$ of length 3 exists. Then, from the condition (2), there exists a vertex $w\neq y$ such that $x,$ $w,$ $z,$ $v$ is a path. Since $c_{3}=3,$ $w$ must coincide with $p$ or $q$ . We may assume that $w=p$ . Then, the condition (2) between $v$ and $p$ shows that $z$ is in $T$ and adjacent to $p$ . From the condition (2) between $z$ and $x,$ $z$ must coincide with $T$ the vertex in adjacent with $p$ but nonadjacent with $q$ . Then, $d(q, z)=3$ follows from the condition $a_{1}=a_{2}=0$ . Now $p,$ $r,$ $v,$ $y$ are adjacent with $z$ and at distance 2 $v$ $p\neq y$ from $q$ . Since $c_{3}=3,$ $y$ must coincide with one of $p,$ $r,$ , but and $d(v,x)=3$ holds, $y=r$ follows; that is, $x$ is adjacent with $p,$ $q,$ $r$ . $\blacksquare$ $3^{*}$ $d_{\Gamma}$ Lemmas 3 and show how determines the parameters $a_{i},$ $c_{i}$ . Conversely, the $d_{\Gamma}$ parameters $a_{i},$ $c_{i}$ of course determine the . 56 $\uparrow LA1$ $n^{-}ith$ FORM( . Let $G$ be $c\prime 1l1$ r-regular distance-regular grapli paiameteis $a_{1}=$ $c\iota_{2}=0$ an $dc_{2}=2,$ $c_{3}=3$ , other $tl_{lal2}H(r)$ . Then, the $t$ in Lemmas 3 and $3^{*}$ is the minimum number $ksuch$ that either $a_{k}\neq 0$ or $c_{k}\neq k$ occurs, and we have $d=\{\begin{array}{l}2t+1ifc_{t}=t2totherwise\end{array}$ PROOF: By Lemma 4, $G$ is a LHDRG. Then, this formula follows from 3 and 3*. $\blacksquare$ \S 4. Multiply transitive groups. Although the complete classification of designs is not accomplished yet, a similar great problem was settled as a result of the classification of finite simple groups; that is, the classification of multiply transitive groups. A permutation group of degree $r$ is said to be $k- t\tau ansitive$ if the induced action on the set of the ordered k-tuples of the element of $X$ is transitive, and said to be k-homogeneous if the induced action on the set of the unordered k-tuples (i.e., on $(_{k}^{X})$ ) is transitive. Even 2-transitive permutation groups were $classified[4][7]$ . In this paper we shall use the following two group-theoretical theorems. THEOREM 1. All k-transitive permutation groups of degree $r$ with $k\geq 6$ are $A_{r}$ and $S_{r}$ . All k-transitive permutation groups with $k=5,4$ except $\mathcal{A}_{r}$ and $S_{r}$ are four $1\backslash Iatl_{1}ieu$ groups $1II_{24},1tI_{12},$ $M_{23},andM_{11}$ . The subscript denotes the degree of the permutation group, and the former two are 5-transitive and the latter two are 4-transitive. THEOREM 2. A k-homogeneous group is $(k-1)$ -transitive. For $k\geq 5$ , k-homogeneous group is k-transitive. There exist only five 4-Aomogeneous but not 4-transitive groups of degre $e$ more than 5; $viz$ . $PSL_{2}(5),$ $PGL_{2}(5),$ $PGL_{2}(8),$ $P\Gamma L_{2}(8)$ , and $P\Gamma L_{2}(32)$ with degree 6, 6, 9, 9, 33 respectively. Theorem 1 is a consequence of the classification of finite simple group. Theorem 2 is proved in [6]. To connect the automorphism group with the fundamental group of $G$ , following lemma is crucial. LEMMA 5. Let $G=H(r)/\Gamma$ be a locally Hamming graph. Then, $Aut(G)\cong N(\Gamma)/\Gamma$ holds, where $N(\Gamma)$ denotes the normalizer of $\Gamma\in Aut(H(r))$ . PROOF: Let $f$ : $H(r)arrow G=H(r)/\Gamma$ be the can\‘onical covering. For an automor- $\delta\Gamma\delta^{-1}=\Gamma$ phism 6 : $H(r)arrow H(r)$ with , we define $\delta_{/\Gamma}$ : $H(r)/\Gammaarrow H(r)/\Gamma$ by $\Gamma x\mapsto\Gamma\delta x$ . It is easy to prove that this mapping is a well-defined automorphism, by $\Gamma\delta x=\delta\Gamma x$ $\delta$ using . The map $N(\Gamma)arrow Aut(H(r)/\Gamma)$ defined by }$arrow\delta_{/\Gamma}$ is obviously a homomorphism of groups. This map is proved to be surjective as follows. Take any 57 $\alpha\in Aut(H(t)/\Gamma)$ . Then, the universality of $f$ asserts that there exists an automor- $\delta$ $\alpha\Gamma x=\Gamma\delta x$ phism : $H(r)arrow H(r)$ such that $\alpha f=f\delta$ holds. This implies for any $x$ . $\delta$ Thus, if is proved to be contained in $N(\Gamma)$ , then clearly $\alpha=\delta_{/\Gamma}$ holds and the sur- jectivity follows, but for any $\gamma\in\Gamma$ , we have $f\delta\gamma\delta^{-1}=\alpha f\gamma\delta^{-1}=\alpha fS^{-1}=f\delta\delta^{-1}=f$ by definition of I“, so $\delta\gamma\delta^{-1}\in\Gamma$ holds; in other words, $\delta\in N(\Gamma)$ holds. Thus we have a surjective homomorphism $N(\Gamma)arrow Aut(H(r)/\Gamma)$ . It remains to prove that the ker- nel of this homomorphism is F. Take any $\delta\in N(\Gamma)$ such that $\delta_{/\Gamma}=id$ . This implies $\Gamma u=\Gamma\delta u$ for all $u$ ; in other words, $fu=f\delta u$ for any $u$ , and $\delta\in\Gamma$ follows from the $\gamma\in\Gamma,$ $\Gamma x$ $\Gamma\gamma x=\Gamma x$ definition of F. Conversely, for any $\gamma/\Gamma$ maps any vertex to , and $\Gamma$ is proved to be the kernel. $\blacksquare$ LEMMA 6. $Aut(H(r))\cong S_{r}.Wr.\mathbb{P}_{2}$ holds. PROOF: Recall that the wreath product $S_{r}.Wr.\ovalbox{\tt\small REJECT}_{2}$ is the set $\{(\sigma, d)|\sigma\in S_{r}, d\in \mathbb{I}_{2}=r\}$ with multiplication $(\sigma, d)\cdot(\sigma’, d’)=(\sigma\sigma’, \sigma d’)$ , where $\sigma\in S_{r}$ is considered as a permutation matrix over $\mathbb{I}_{2}^{=}$ . We correspond $(\sigma, d)$ to a mapping $H(r)arrow H(r)$ defined by $z\mapsto\sigma z+d$ . This defines the above isomorphism. Injectivity and surjectivity are easily checked by using Proposition 1. $\blacksquare$ From now on we deal with only locally Hamming distance-transitive graphs with discreteness $d$ and valency r(denoted by $LHDTG_{d}^{r}$ ). Next lemma connects LHDTG with multiply transitive group. LEMMA 7. Let $G\cong H(r)/\Gamma$ be a graph contain$ed$ in $LHDTG_{d}^{r}$ . Define the point stabilizer $N(\Gamma)_{\emptyset}$ $:=\{\delta\in N(\Gamma)|\delta\emptyset=\emptyset\}$ . Then, $N(\Gamma)_{\emptyset}$ can be regarded as a subgroup of $N(\Gamma)/\Gamma\cong Aut(G)$ . Also, $N(\Gamma)_{\emptyset}$ can be regarded as a subgroup of $S_{r}$ , and is a k-homogeneous group acting on the r-element subset $X$ for $k=\lfloor(d-1)/2\rfloor$ . Moreover, $N(\Gamma)_{\emptyset}$ acts on the block set of the design defined in Lemmas 3 and 3*; $i.e.$ , on $\Gamma\emptyset\cap(_{d}^{X})$ . PROOF: Case $d=2t+1$ . In this case we have $k=t$ . Take vertices $u,$ $v\in H(r)$ such that $d(\emptyset, u)=d(\emptyset, v)=t$ holds. Since $N_{t}(\emptyset)\cong N_{t}(\Gamma\emptyset)$ holds, $d_{G}(\Gamma\emptyset, \Gamma u)=d_{G}(\Gamma\emptyset, \Gamma v)=t$ holds. Since $H(r)/\Gamma$ is distance-transitive, there exists a $\delta_{/\Gamma}\in Aut(H(r)/\Gamma)$ such that $\delta_{/\Gamma}(\Gamma\emptyset)=\Gamma\emptyset$ and $\delta_{/\Gamma}(\Gamma u)=\Gamma v$ . We can take a $\delta\in N(\Gamma)$ such that $\delta_{/\Gamma}$ coincides with the one defined in Lemma 4 by the surjectivity of $\delta\mapsto\delta_{/\Gamma}$ . Since $\Gamma\delta\emptyset=\delta_{/\Gamma}(\Gamma\emptyset)=\Gamma\emptyset$ $\delta$ $\delta\emptyset=\emptyset$ holds, a $\gamma\in\Gamma$ satisfies $\gamma\delta\emptyset=\emptyset$ , and by retaking $\gamma\delta$ as , we may assume that : i.e., $\delta\in N(\Gamma)_{\emptyset}$ . Now $\Gamma\delta u=\Gamma v$ holds, and since $N_{t}(\emptyset)\cong N_{t}(\Gamma\emptyset),$ $\delta n=v$ holds; i.e., 58 $N(\Gamma)_{\emptyset}$ acts on $X$ t-homogeneously. Since $\Gamma$ has no fixed point, $\Gamma\cap N(\Gamma)_{\emptyset}=$ {id} holds, and this implies that $N(\Gamma)_{\emptyset}c$ an be embedded into $N(\Gamma)/\Gamma$ as a subgroup. On the other hand, $N(\Gamma)_{\emptyset}cal)$ be naturally identified with a subgroup of $S_{r}$ , since through the identifi $c$ ation $Aut(H(r))\cong S_{r}.Wr.F_{2}$ , we have $N(\Gamma)_{\emptyset}\subset\{(\sigma,0)|\sigma\in S_{r}\}\cong S_{r}$ . Take an element $x=\gamma\emptyset\in\Gamma\emptyset\cap(_{2t}^{X})$ and an element $g\in N(\Gamma)_{\emptyset}$ . Then, $g=(\sigma, 0)$ and $\sigma x=gx=g\gamma\emptyset=\gamma’g\emptyset=\gamma’\emptyset\in\Gamma\emptyset\cap(_{2t+1}X)$ holds for some $\gamma’\in\Gamma$ since $g\in N(\Gamma)$ . Thus, we have $gx\in\Gamma\emptyset\cap(_{2t+1}X)$ , and consequently, $N(\Gamma)_{\emptyset}$ acts on the blo $ck$ set $\Gamma\emptyset\cap(_{2t+1}X)$ . For $d=2t$ , the same proof is also valid with modification on only the parameters $t$ and $d$ . $\blacksquare$ PROOF OF MAIN RESULT: Using Formula 1, we see that the discreteness $d$ is calcu- lated as shown in Main $Re$sult. Then, all statements follow from Lemmas 3,3*, and 7, and Criterion 1. $\blacksquare$ LEMMA 8. Let $G=H(r)/\Gamma,$ $N(\Gamma)_{\emptyset}$ be as in Lemma 7. If $N(\Gamma)_{\emptyset}$ is isomorphic to $A_{r}$ or $S_{r}$ , then $G$ is a Hamming scheme or a folded Hamming scheme. $\Gamma$ PROOF: Let $d_{\Gamma}$ be the discreteness of , and let Z3 be the blo $ck$ set of the $t-(r, d, \lambda)$ design stat $ed$ in Lemma 7. If $d_{\Gamma}=\infty$ then $G=H(r)$ , thus we may assume $d_{\Gamma}<\infty$ . Take a block $B\in 23$ . Since any cyclic permutation of length 3 is contained in both $\mathcal{A}_{r}$ and $S_{r}$ , there exists a $\gamma\in\Gamma$ with $d(B,\gamma B)\leq 2$ , unless $B=X$ holds. Since $d_{\Gamma}\geq 5$ , $B=X$ must hold. Then, $d_{\Gamma}=r$ , and the fiber on the vertex $\Gamma\emptyset$ contains exactly two vertices $\emptyset,$ $X\in V(H(r))$ . A similar situation occurs on any vertex on $G$ , an $d$ consequently, $G$ is obtained from $H(r)$ by identifying two antipodal vertices; i.e., a folded Hamming scheme. $\blacksquare$ COROLLARY OF LEMMA 8. All graph $s$ coniain$ed$ in $LHDTG_{d}^{r}$ for $d\geq 13$ are the Hamming schemes or the folded Hamming schemes. PROOF: For $d\geq 13$ , either $d=2t+1$ with $t\geq 6$ or $d=2t$ with $t\geq 7$ holds. In each case, $N(\Gamma)_{\emptyset}$ is at least 6-homogeneous, and the result follows from Theorems 1 and 2 and Lemmas 7 and 8. $\blacksquare$ Combining a calculation on the admissibility of parameters of a design and Crite- $d$ rion 1, we $c$ an improve the lower bound of . LEMMA 9. Corollary of $Lem$ma 8 holds for $d\geq 9$ . PROOF: Since for any distinct $B,$ $B’\in B\#(B\cap B’)\leq t$ holds, an inequality $2d-t\leq r$ must hold. Thus, $18\leq 2d\leq r$ follows, an $d$ the degree of the corresponding homoge- neous group is no less than 18. There are three 4-homogeneous groups satisfying this condition except $\mathcal{A}_{r}$ and $S_{r}$ ; viz, $P\Gamma L_{2}(32),$ $M_{23}$ , and $M_{24}$ . Let $G=H(r)/\Gamma$ be a graph in $LHDTG_{d}^{r}$ which is neither a Hamming. nor a folded Hamming $s$ cheme. The next table lists all $\lfloor(d-1)/2\rfloor$ -homogeneous groups for $d_{\Gamma}=$ $10,11,12$ . The number $s$ in the column homogeneousity shows that the group $N(\Gamma)_{\emptyset}$ is at least s-homogeneous; i.e., $s=\lfloor(d-1)/2\rfloor$ , and the column $N(\Gamma)_{\emptyset}$ lists all the 59 s-homogeneous groups. The column $ad$ missibility shows the $necessa1^{\cdot}y$ condition $on$ $/\backslash deduced$ from the well-known admissibility condition of a $t-(r, d, /\backslash )$ design; that is, $(_{t-i}^{d-i})|/\backslash (tr--ii)$ , where $a|b$ denotes that $a$ divides $b$ . $d_{\Gamma}t$ Homogeneousity $N(\Gamma)_{\emptyset}$ $r=\#(X)$ Criterion 1 Admissibility 126 5-homogeneous $M_{24}$ 24 $/\backslash \leq 3$ $42|\lambda$ 115 5-homogeneous $M_{24}$ 24 $/\backslash \leq 19/6$ $42|\lambda$ 105 4-homogeneous $M_{24}$ 24 $/\backslash \leq 19/5$ $18|\lambda$ $M_{23}$ 23 $\lambda\leq 18/5$ $252|/\backslash$ $P\Gamma L_{2}(32)$ 33 $\lambda\leq 28/5$ $504|/\backslash$ 9 4 4-homogeneous $M_{24}$ 24 $\lambda\leq 4$ $24|/\backslash$ $M_{23}$ 23 $/\backslash \leq 19/5$ $18|/\backslash$ $P\Gamma L_{2}(32)$ 33 $\lambda\leq 29/5$ $21|/\backslash$ List 1. Possible parameters for $LHDTG_{d}^{r},$ $9\leq d\leq 12$ . In each case, Criterion 1 contradicts the admissibility. $\blacksquare$ THEOREM 3. Let $G$ be a distance-transitive graph with paramet$ersa_{i}=0$ for $i=$ $1,2,3$ and $c_{i}=i$ for $i=1,2,3,4$ . Then, $G$ is a Hamming scheme or a folded Hamming scheme. PROOF: Formula 1 implies $t\geq 4$ and $d\geq 9$ . Thus, this theorem is a direct conse- quence of Lemma 9. $\blacksquare$ \S 5. Nontrivial Examples. The next table lists some known locally Hamming distance-regular graphs other than Hamming scheme and folded Hamming scheme. All these examples are cited from [3, pp.480-483]. The column Intersection Array shows the intersection array $\{b_{0}, b_{1}, \ldots , b_{D-1} ; c_{1}, c_{2}, \ldots c_{D}\}$ , where $D$ denotes the diameter. After the array is the reference to [3]. $The*in$ the column $N(\Gamma)_{\emptyset}$ indicates that it is not distance-transitive. No. $d_{\Gamma}tr$ $N(\Gamma)_{\emptyset}\#(V(G))$ Intersection Array 1 8 424 $M_{24}$ 4096 {24, 23, 22, 21; 1, 2, 3, 24}, Ch.11.3.2 2 8 423 $M_{23}$ 4096 {23, 22, 21, 20, 3, 2, 1; 1, 2, 3, 20, 21, 22, 23}, p.362 $M_{23,*}$ 2048 {23, 22, 21; 1, 2, 3}, Ch.11.3.4 $43$ $77322323$ 2048 {22, 21, 20, 3, 2, 1; 1, 2, 3, 20, 21, 22}, p.365 5 6 322 $M_{22}$ 2048 {22, 21, 20, 16, 6, 2, 1; 1, 2, 6, 16, 20, 21, 22}, p.363 $M_{22}*$ 1024 {22, 21, 20; 1, 2, 6}, Ch.11.3.5 $76$ $56221322$ 1024 {21, 20, 16, 6, 2, 1; 1, 2, 6, 16, 20, 21}, p.365 8 5 221 $*$ 2048 {21, 20, 16, 9, 2, 1; 1, 2, 3, 16, 20, 21}, p.365 9 5 221 $P\Gamma L_{3}(4)$ 512 {21, 20, 16; 1, 2, 12}, Ch.11.3.6 List 2. Some known $LHDRG_{d}^{r}$ with $5\leq d\leq 8$ . 60 For No.1-6, we shall briefly describe the corresponding designs. For No.1, the corre- spondi $ng$ is a 4-(24,8,5) design, which coincides with the block set of the Witt system $S(5,8,24)$ . To No.2, a 4-(23,8,4) design corresponds, which has the same number of blocks with the Witt system $S(4,7,23)$ . This design is obtained by adding one element to each block in $S(4,7,23)$ so that $M_{23}$ acts on those blocks. To No.3, a 3-(23,7,5) design corresponds, which coincides with the block set of the Witt system $S(4,7,23)$ . To No.4, a 3-(22,7,4) design corresponds, which has $2^{4}\cdot 11$ blocks. The author doesn’t know how to obtain this design from $S(3,6,22)$ . To No.5 and No.6, a 3-(22,6,1) design, in other words, the Witt system $S(3,6,22)$ corresponds. This implies that the design does not determi $ne$ the $gr$ aph uniquely. All of the listed graphs $are$ obtained as a coset graph of modified binary Golay $3$ $codes$ [ , Ch.11.3]. Its fundamental group is a linear subspace in $F_{2}^{r}$ when regarded as a subgroup of $S_{r}\rangle\triangleleft\#_{2}^{=r}$ . This invokes the next conjecture. CONJECTURE 1. Let $\Gamma\subset Aut(H(r))$ be a subgroup with discreteness at least 5. If $\Gamma$ $1’$ $H(r)/\Gamma$ is distance-regular, then is a linear subspace in $\#_{2}^{=r}$ through the dentification $S_{r}*\#_{2}=r$ . CONJECTURE 2. Under the same assumption, $\Gamma$ is a subset of a linear code obtained $sl1$ from bin$ary$ Golay codes by truncation or ortenin$g$. Conjecture 2 implies the complete classification of LHDRG. To prove or disprove these conjectures does not seem to be too difficult, at leas $t$ for the distance-transitive $c$ ase with $d_{\Gamma}\geq 7$ ; i.e., with $a_{1}=a_{2}=0,$ $c_{2}=2,$ $c_{3}=3$ , because in this case the point stabilizer is a 2-homogeneous group $on$ which we have much information[7]. $b_{i},$ In this paper, we have not utilized any information on the parameters $a_{i},$ $c_{i}$ for $i\geq t+1$ , so the same proof $c$ an also be applied to graphs for whi $ch$ the parameters with subscript $i>tc$an not be defined. Also, we have not used the properties of association schemes at all, so it seems to be possible that one proves a stronger result with easier proof than this paper, using such structures. REFERENCES 1. Bannai, E. and Ito, T., (Algebraic Combinatorics I,” Benjamin, 1984. 2. Brouwer, A. E., On the uniqueness of a regular thin near octagon (or partial 2-geometry, or parallelism) derived from the binary Goiay code, IEEE Trans. Inf. Theory 29 (1987), 370-371. ) 3. Brouwer, A.E. et. al., “Distance-Regular $Graph_{S},$ Ergebnisse der Mathematik und ihrer Gren- zgebiete, 3.Folge.Band 18, Springer, 1989. 4. Cameron, P.J., Finite permutation groups and finite simple $g\tau oups$ , Bull. London. Math. Soc. 13 (1981), 1-22. 5. Egawa, Y., Characterization of $H(n,$q) by the parameters, J. Combin. Th. (A) 31 (1981), 108-125. 6. Kantor, W.M., 4-Homogeneous Groups, Math. Zeitschr. 103 (1968), 67-68. 7. , W.M., Homogeneous designs and geometric lattices, J. Combin. Th. 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